In a statement posted on X, formerly Twitter, Mrs Obama called her mother her "rock, always there for whatever I needed".
"She was the same steady backstop for our entire family, and we are heartbroken to share she passed away today," she wrote.
In a separate tweet, Mr Obama said that “there was and will be only one Marian Robinson”.
“In our sadness, we are lifted up by the extraordinary gift of her life,” he added. “And we will spend the rest of ours trying to live up to her example.”
No further details were given the cause of death.
Born in 1937, Robinson grew up one of seven children in Chicago, the city where she spent much of her life before agreeing to come to Washington DC after Mr Obama’s electoral victory.
Early in her life, she studied to become a teacher before working as a secretary. She raised Michelle and her other child, Craig, together with her husband Frasier Robinson on Chicago's South Side.
"At every step, as our families went down paths none of us could have predicted, she remained our refuge from the storm," the Obama statement said.
"On Election Night in 2008, when the news broke that Barack would soon shoulder the weight of the world, she was there, holding his hand."
An image taken on the night in 2008 when her son-in-law made history as the nation's first African-American president showed Robinson sitting on a sofa with him, watching the results come in.
The statement added that Robinson had agreed to move to the White House after a “healthy nudge” from Barack and Michelle Obama, who, along with their daughters, “needed her”.
She later spoke of how she insisted on doing her own laundry there.
In an interview with CBS, the BBC’s US partner, Robinson said she had felt compelled to move to Washington because she felt “like this was going to be a very hard life” for her daughter and son-in-law.
“And I was worried their safety,” she added. “I was worried my grandkids. That’s what got me to move to DC.”
The lifelong Chicago resident had never boarded a flight out of the US until she flew aboard Air Force One with the Obamas to France in 2009.
Robinson - whom Mr Obama once called “the least pretentious person” he knew - said that it was a “huge adjustment” to have her needs met by White House staff.
"Rather than hobnobbing with Oscar winners or Nobel laureates, she preferred spending her time upstairs with a TV tray, in the room outside her bedroom with big windows that looked out at the Washington Monument," the family statement said.
"The only guest she made a point of asking to meet was the Pope," it added.
